5CAT(0) spaces and groups
IV Topics in Geometric Group Theory
5.3 Length metrics
In differential geometry, if we have a covering space
˜
X → X
, and we have a
Riemannian metric on
X
, then we can lift this to a Riemannian metric to
˜
X
.
This is possible since the Riemannian metric is a purely local notion, and hence
we can lift it locally. Can we do the same for metric spaces?
Recall that if γ : [a, b] → X is a path, then
`(γ) = sup
a=t
0
<t
1
<···<t
n
=b
n
X
i=1
d(γ(t
i−1
), γ(t
i
)).
We say γ is rectifiable if `(γ) < ∞.
Definition
(Length space)
.
A metric space
X
is called a length space if for all
x, y ∈ X, we have
d(x, y) = inf
γ:x→y
`(γ).
Given any metric space (
X, d
), we can construct a length pseudometric
ˆ
d : X × X → [0, ∞] given by
ˆ
d(x, y) = inf
γ:x→y
`(γ).
Now given a covering space
p
:
˜
X → X
and (
X, d
) a metric space, we can define
a length pseudometric on
˜
X by, for any path ˜γ : [a, b] →
˜
X,
`(˜γ) = `(p ◦ ˜γ).
This induces an induced pseudometric on
˜
X.
Exercise. If X is a length space, then so is
˜
X.
Note that if
A, B
are length spaces, and
X
=
A ∪ B
(such that the metrics
agree on the intersection), then
X
has a natural induced length metric. Recall we
previously stated the Hopf–Rinow theorem in the context of differential geometry.
In fact, it is actually just a statement about length spaces.
Theorem
(Hopf–Rinow theorem)
.
If a length space
X
is complete and locally
compact, then X is proper and geodesic.
This is another application of the Arzel´a–Ascoli theorem.